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Understanding Discrete Random Variables: Definitions, PMFs, and Key Properties

This chapter explores the concept of discrete random variables (RVs), including their definitions, probability mass functions (PMFs), and key properties. It delves into various families of discrete RVs like Bernoulli, Geometric, Binomial, and Poisson distributions, highlighting their PMFs and expected values. Additionally, it covers cumulative distribution functions (CDFs), averages, variance, moments, and conditional PMFs. MATLAB applications are also discussed for practical understanding of these concepts.

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Understanding Discrete Random Variables: Definitions, PMFs, and Key Properties

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  1. Chapter 2pp. 49-100 William J. Pervin The University of Texas at Dallas Richardson, Texas 75083

  2. Chapter 2 Discrete Random Variables

  3. Chapter 2 2.1 Definitions: A random variable (X) consists of a experiment with a probability measure P[.] defined on a sample space S and a function X that assigns a real number X(s) to each outcome sS.

  4. Chapter 2 Shorthand notation: {X=x} ≡ {s  S | X(s) = x} Discrete vs. Continuous RVs

  5. Chapter 2 2.2 Probability Mass Function: The PMF (PX) of the discrete random variable X is PX(x) = P[X=x] = P[{s  S | X(s) = x}]

  6. Chapter 2 Theorem: For any discrete random variable X with PMF PX and range SX: • (x) PX (x) ≥ 0 • ΣxSX PX(x) = 1 • (BSX) P[X  B] = P[B] = ΣxB PX(x)

  7. Chapter 2 2.3 Families of Discrete RVs Bernoulli (p) RV: (0 < p < 1) PX(x) = 1-p if x=0, p if x=1, 0 otherwise (Two outcomes)

  8. Chapter 2 Geometric (p) RV: (0 < p < 1) PX(x) = (1-p)x-1p, x=1,2,…; 0 otherwise (Number to first success) Binomial (n,p) RV: (0 < p < 1; n = 1,2,…) PX(x) = C(n,x)px(1-p)n-x (Number of successes in n trials) (Note: Binomial(1,p) is Bernoulli)

  9. Chapter 2 Pascal (n,p) RV: (0 < p < 1; n = 1,2,…) PX(x) = C(x-1,n-1)pk(1-p)x-n (Number to n successes) (Note: Pascal(1,p) is Geometric) Discrete Uniform (m,n) RV: (m<n integers) PX(x) = 1/(n-m+1) for x=m,m+1,…,n; 0 otherwise

  10. Chapter 2 Poisson (α) RV: (α > 0) PX(x) = αxe-α /x! for x=0,1,…; 0 otherwise (Arrivals: α = λT)

  11. Chapter 2 2.4 Cumulative Distribution Function The CDF (FX) of a random variable X is FX(x) = P[X ≤ x]

  12. Chapter 2 For any discrete random variable X with range SX = {x1 ≤ x2 ≤ …}: the CDF (FX) is monotone non-decreasing from 0 to 1, with jump discontinuities of height PX(xi) at each xi SX and constant between the jumps.

  13. Chapter 2 FX(b) – FX(a) = P[a < X ≤ b]

  14. Chapter 2 2.5 Averages Statistics: mean, median, mode, … Parameter of a model: mode, median Expected Value of X = E[X] = μX = ΣxSX xPX(x)

  15. Chapter 2 E[X] = p if X is Bernoulli (p) RV E[X] = 1/p if X is geometric (p) RV E[X] = α if X is Poisson (α) RV E[X] = np if X is binomial (n,p) RV E[X] = k/p if X is Pascal (k,p) RV E[X] = (m+n)/2 if X is discrete uniform (m,n) RV

  16. Chapter 2 Note: Poisson PMF is limiting case of binomial PMF.

  17. Chapter 2 2.6 Functions of a Random Variable Derived RV Y = g(X) for RVs when y = g(x) for values PY(y) = Σx:g(x)=y PX(x)

  18. Chapter 2 2.7 Expected Value of a Derived RV If Y = g(X) then E[Y] = μY = ΣxSX g(x)PX(x) For any RV X: E[X-μX] = 0 and E[aX + b] = aE[X] + b

  19. Chapter 2 E[X2] = ΣxSX x2 P(x)

  20. Chapter 2 2.8 Variance and Standard Deviation Var[X] = E[(X-μX)2] σX = sqrt(Var[X]) Var[X] = E[X2] – (E[X])2= E[X2] – μX2

  21. Chapter 2 Moments of a RV X: nth moment: E[Xn] nth central moment: E[(x – μX)n] Theorem: Var[aX + b] = a2 Var[X]

  22. Chapter 2 Var[X] = p(1-p) if X is Bernoulli (p) RV Var[X] = (1-p)/p2 if X is geometric (p) RV Var[X] = α if X is Poisson (α) RV Var[X] = np(1-p) if X is binomial (n,p) RV Var[X] = k(1-p)/p2 if X is Pascal (k,p) RV Var[X] = (n-m)(n-m+2)/12 if X is discrete uniform (m,n) RV

  23. Chapter 2 2.9 Conditional PMF PX|B(x) = P[X=x|B] 2.10 MATLAB

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